Grade 9 · Free math lesson
Quadratic Equations
After this lesson, you will be able to solve simple quadratic equations and understand what their solutions mean.
Practice Quadratic Equations freeSkills in this unit
Pick a skill to try three free questions on just that.
- Multiplying binomials
- Factoring trinomials
- Difference of squares and perfect square trinomials
- Solving quadratics by factoring
- The quadratic formula
- Completing the square
- Vertex, axis of symmetry, and graphing parabolas
- The discriminant and the number of solutions
- Quadratic word problems
What Is a Quadratic?
A quadratic equation always has an x² in it — that's what makes it special. It looks like ax² + bx + c = 0, where a, b, and c are just numbers.
Example: x² + 5x + 6 = 0 is a quadratic. x + 5 = 0 is NOT — it has no x².
Remember: Spot the x² and you've spotted a quadratic!
Factor to Find Answers
Factoring means splitting the equation into two brackets that multiply together to make zero. If two things multiply to give zero, at least one of them must BE zero.
Example: x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0. So x = -2 or x = -3.
Remember: Find two numbers that MULTIPLY to c and ADD to b — those go in your brackets!
Solutions on a Graph
The solutions of a quadratic are the spots where its U-shaped curve (called a parabola) crosses the x-axis. There can be 0, 1, or 2 crossing points.
Example: For x² + 5x + 6 = 0, the parabola crosses the x-axis at x = -2 and x = -3. Those are your two answers!
Remember: Solutions = where the curve LANDS on the x-axis.
Worked example
Solve x² + 7x + 12 = 0
- 1Step 1: Find two numbers that multiply to 12 AND add to 7. That's 3 and 4 (3 × 4 = 12, 3 + 4 = 7).
- 2Step 2: Write the factored form — (x + 3)(x + 4) = 0.
- 3Step 3: Set each bracket equal to zero — x + 3 = 0 gives x = -3, and x + 4 = 0 gives x = -4.
Answer: x = -3 or x = -4
The #1 mistake to avoid
Students forget to flip the sign! If your bracket is (x + 3) = 0, then x = -3, NOT +3. Always move the number to the other side of the equals sign.
Key takeaways
- A quadratic equation always contains x² and equals zero.
- You can solve it by factoring — find two numbers that multiply to c and add to b.
- The solutions are where the parabola crosses the x-axis — there can be 0, 1, or 2 of them.
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